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x cos^(-1)x |
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Answer» Solution :`" Let " I = INT x cos^(-1) x dx` `underset(RARR x = cot t rArr dx=- sin t dt)("Let " cos^(-1) x=t)` ` :. I= int x cos^(-1) xdx =- int t cos t. sin t dt` `=-(1)/(2) int t.2 sin t cot t dt=- (1)/(2) int t . sin 2t dt` ` I= -(1)/(2) [t int sin 2t dt - int {(d)/(dx) (t). int sin 2t dt }dt]` ` =-(1)/(2) [t((-cos 2t))/(2)+ int 1. (cos 2t)/(2)dt]` `=(1)/(2) t cos 2t -(1)/(4)(sin 2t)/(2)+C` ` =(1)/(4)t cos 2t - (1)/(8) sin 2t +C` `=(1)/(4) t (2cos^(2) t-1)-(1)/(8) sin 2t+C` ` =(1)/(4) t (2 cos^(2) t-1)- (1)/(4) (1-cos^(2)t)^(1//2) cos t +C` ` :. intx cos^(-1) x dx =(1)/(4) cos^(-1) x (2x^(2)-1)` `- (1)/(4)(1-x^(2))^(1//2) x+C` `=(1)/(2) (2x^(2) -1) cos^(-1) x-(1)/(4) x SQRT(1-x^(2)) +C` |
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